FQHE and $t{t}^{\ast }$ geometry

Abstract

Cumrun Vafa has proposed a microscopic description of the Fractional Quantum Hall Effect (FQHE) in terms of a many-body Hamiltonian $H$ invariant under four supersymmetries. The non-Abelian statistics of the defects (quasi-holes and quasi-particles) is then determined by the monodromy representation of the associated $t{t}^{\ast }$ geometry. In this paper we study the monodromy representation of the Vafa 4-susy model. Modulo some plausible assumption, we find that the monodromy representation factors through a Temperley-Lieb/Hecke algebra with $q=\pm \exp (\pi i/\nu )$. The emerging picture agrees with the other Vafa's predictions as well. The bulk of the paper is dedicated to the development of new concepts, ideas, and techniques in $t{t}^{\ast }$ geometry which are of independent interest. We present several examples of these geometric structures in various contexts.

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• A slow review of the AGT correspondence
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